Integrand size = 27, antiderivative size = 105 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=-\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac {1}{2} d^2 e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+2 d^2 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
-(-e^2*x^2+d^2)^(3/2)/x-1/2*d^2*e*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+2*d^2*e *arctanh((-e^2*x^2+d^2)^(1/2)/d)-1/2*e*(e*x+4*d)*(-e^2*x^2+d^2)^(1/2)
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^2-4 d e x+e^2 x^2\right )}{2 x}+d^2 e \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+2 d \sqrt {d^2} e \log (x)-2 d \sqrt {d^2} e \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
(Sqrt[d^2 - e^2*x^2]*(-2*d^2 - 4*d*e*x + e^2*x^2))/(2*x) + d^2*e*ArcTan[(e *x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] + 2*d*Sqrt[d^2]*e*Log[x] - 2*d*Sqrt [d^2]*e*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]]
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {570, 540, 27, 535, 538, 224, 216, 243, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {\int \frac {d^2 e (2 d+e x) \sqrt {d^2-e^2 x^2}}{x}dx}{d^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -e \int \frac {(2 d+e x) \sqrt {d^2-e^2 x^2}}{x}dx-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle -e \left (\frac {1}{2} d^2 \int \frac {4 d+e x}{x \sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} (4 d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -e \left (\frac {1}{2} d^2 \left (e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+4 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx\right )+\frac {1}{2} (4 d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -e \left (\frac {1}{2} d^2 \left (4 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} (4 d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -e \left (\frac {1}{2} d^2 \left (4 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (4 d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -e \left (\frac {1}{2} d^2 \left (2 d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2+\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {1}{2} (4 d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -e \left (\frac {1}{2} d^2 \left (\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {4 d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}\right )+\frac {1}{2} (4 d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -e \left (\frac {1}{2} d^2 \left (\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+\frac {1}{2} (4 d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}\) |
-((d^2 - e^2*x^2)^(3/2)/x) - e*(((4*d + e*x)*Sqrt[d^2 - e^2*x^2])/2 + (d^2 *(ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]))/2 )
3.2.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p + 1) Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Time = 0.43 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(-\frac {d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{x}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x}{2}-\frac {e^{2} d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}+\frac {2 e \,d^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}-2 e d \sqrt {-e^{2} x^{2}+d^{2}}\) | \(140\) |
| default | \(\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}}{d^{2}}-\frac {2 e \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{d^{3}}+\frac {\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}}{d^{2}}+\frac {2 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{d^{3}}\) | \(674\) |
-d^2*(-e^2*x^2+d^2)^(1/2)/x+1/2*(-e^2*x^2+d^2)^(1/2)*e^2*x-1/2*e^2*d^2/(e^ 2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+2*e*d^3/(d^2)^(1/2)*ln ((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-2*e*d*(-e^2*x^2+d^2)^(1/2)
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\frac {2 \, d^{2} e x \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 4 \, d^{2} e x \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 4 \, d^{2} e x + {\left (e^{2} x^{2} - 4 \, d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, x} \]
1/2*(2*d^2*e*x*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 4*d^2*e*x*log(- (d - sqrt(-e^2*x^2 + d^2))/x) - 4*d^2*e*x + (e^2*x^2 - 4*d*e*x - 2*d^2)*sq rt(-e^2*x^2 + d^2))/x
Result contains complex when optimal does not.
Time = 3.48 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.14 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=d^{2} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) \]
d**2*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e **2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-d/(x*sqrt (1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2) ), True)) - 2*d*e*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*aco sh(d/(e*x)) - e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*d*asin(d/(e*x)) + I*e*x/sqr t(-d**2/(e**2*x**2) + 1), True)) + e**2*Piecewise((d**2*Piecewise((log(-2* e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/2 + x*sqrt(d**2 - e**2*x**2)/2, Ne(e**2 , 0)), (x*sqrt(d**2), True))
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.20 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=-\frac {d^{2} e^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} + 2 \, d^{2} e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{2} x - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d e - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{x} \]
-1/2*d^2*e^2*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 2*d^2*e*log(2*d^2/abs (x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) + 1/2*sqrt(-e^2*x^2 + d^2)*e^2*x - 2*sqrt(-e^2*x^2 + d^2)*d*e - sqrt(-e^2*x^2 + d^2)*d^2/x
Exception generated. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^2\,{\left (d+e\,x\right )}^2} \,d x \]